An Effective Topological Representation and Dimensionality Reduction Approach for Multi-Material Structural Topology Optimization

Inspired by nature’s multi-material compliant designs and the capability of modern manufacturing methods to make multi-material components without assembly, and motivated by the versatility and enhanced functionality with multi-material designs, we formalize the notion and the design of heterogeneous compliant mechanisms. In the topology optimization of compliant mechanisms, a fictitious density function is often used for material interpolation to overcome the computational difficulties encountered in the large “0-1” type integer programming problem. In this paper, we illustrate that a gradually formed continuous peak function can be used for material interpolation. One of the advantages of introducing the peak function is that multiple materials can easily be incorporated into the topology optimization without increasing the number of design variables. By using the peak function and the optimality criteria method, we synthesize compliant mechanisms with multiple materials with and without the material resource constraint. The numerical examples include the two, three, and four material designs where void is treated as one material. This new design method enables us to optimally juxtapose stiff and flexible materials in heterogeneous compliant mechanisms that can be built using a variety of modern manufacturing methods.

SummaryThis article proposes an efficient approach for solving three‐dimensional (3D) topology optimization problem. In this approach, the number of design variables in optimization as well as the number of degrees of freedom in structural response analysis can be reduced significantly. This is accomplished through the use of scaled boundary finite element method (SBFEM) for structural analysis under the moving morphable component (MMC)‐based topology optimization framework. In the proposed method, accurate response analysis in the boundary region dictates the accuracy of the entire analysis. In this regard, an adaptive refinement scheme is developed where the refined mesh is only used in the boundary region while relating coarse mesh is used away from the boundary. Numerical examples demonstrate that the computational efficiency of 3D topology optimization can be improved effectively by the proposed approach.

A robust topological derivative-based multi-material optimization approach: Optimality condition and computational algorithm

With the rapid advancement of computing technology, mathematical modeling and computer simulations have become common and standard methods for design of complex systems. To exercise these models intelligently and eliminate the burden of manual iteration, manipulating inputs and reviewing outputs, optimization strategies are applied in a simulation-based design environment. In reality, the knowledge that engineers have about the design problem is imperfect and incomplete, uncertainty always exists in simulation-based design. The conventional deterministic optimization techniques do not consider the impact of such uncertainties. If the system responses driving the design decisions are very sensitive to these uncertainties, serious problems might confront the designer. Performance of the actual system may differ greatly from expected optimum performance. Again, if the design solution is near one or more constraint boundaries, then even slight uncertainties or changes in the operating environment could produce failed, unsafe designs, and/or result in substantial performance degradation. In recent years, the concept of reliability and robust design has become very popular. Probabilistic design analysis and statistical modeling allow the identification of designs that qualify as not only feasible, but as consistently feasible in the face of uncertainty. Reliability based topology optimization (RBTO) extends the reliability notion to the area of structural topology optimization. Although reliability based design optimization (RBDO) is a well established research area, very few work has been presented in the area of RBTO. In this study, we give an overview of the various works done in the field of RBTO. Ever since Bendsoe and Kikuchi introduced the topology optimization using a homogenization method, many topology optimization methods have been developed for the both linear and nonlinear structures. Traditional topology optimization methods drive the topology of a structure to an optimum design based on the constraint on mass. Kharmanda et al. introduced reliability constraints in deterministic topology optimization problem. They proposed a heuristic strategy that aims to reduce mass while improving the reliability level of the structure without greatly increasing its weight. But the limit state function used by them was not based on failure criteria for the structure. Their formulation considered uncertainty with respect to ∗Design Automation Laboratory †Advisor geometrical dimension and applied load only. Also their reliability analysis seems to be independent of the boundary and loading condition, so their results showed similar values for the uncertain variables for different structures. Tovar et al. have developed the Hybrid Cellular Automaton (HCA) method for structural synthesis of continuum material where the state of each cell is defined by both density and strain energy. In Agarwal, a decoupled RBDO approach is employed such that the topology optimization is separate from the reliability analysis. Patel et al. showed the use of RBTO using the gradient free Hybrid Cellular Automata (HCA) method. Their formulation incorporates uncertainty with respect to material property also. They considered limit state function based on failure modes on the output displacements. This study aims at giving a comparative study on these methods. It points out the advantages and disadvantages of the methods, and their significance in the field of RBTO. As large numbers of design variables are associated with continuum topology optimization problems, RBTO methods are inherently computationally expensive because of additional required system analysis associate with RBDO. Based on the observations, this study presents an efficient method for incorporating reliability analysis in topology optimization.

Muti-objective Topology Optimization of an Electric Vehicle's Traction Battery Enclosure

Concurrent multi-material and multi-scale design optimization of fiber-reinforced composite material and structures for minimum structural compliance

An approach for the concurrent homogenization-based microstructure type and topology optimization problem

The development of multimaterial topology optimization offers significant advancements in the design of lightweight aerospace structures. A three-field multimaterial topology optimization method with a material selection function under a mass constraint is developed. By integrating a multimaterial linear interpolation model with floating projection techniques, design variables are refined to the expected 0/1 level, ensuring accurate smooth design. The approach demonstrates smooth convergence even when applied to more than three material phases, producing optimized structures with distinct material boundaries and interfaces. A case study of the aircraft engine pylon design is presented, utilizing real material properties, including Poisson’s ratio, to meet lightweight design requirements. The requirement of lightweight design is realized by introducing effective lightweight materials and automatically adjusting the distribution and volume fraction of materials. Additionally, the three-field density representation technique is applied to improve the optimization efficiency and is extended to design considering manufacturing uncertainty. The robust manufacturing multimaterial structure is designed with the potential for manufacturing errors in mind. It can provide valuable guidance for multimaterial lightweight design of aerospace structures.

In this research, a reliability based topology optimization (RBTO) for structural design methodology using the Hybrid Cellular Automata (HCA) method is proposed. More speciflcally, a decoupled reliability based design optimization (RBDO) approach is utilized, so that the topology optimization is separate from the reliability analysis. In this paper, a maximum allowable displacement failure mode is considered. In this methodology, starting from a continuum design space of uniform material distribution and initial uncertain variable values, a deterministic topology optimization is followed by a reliability assessment of the resulting structure to determine the most probable point of failure (MPP) for the current structure. The MPP is determined with respect to the maximum allowable de∞ection of the structure when loaded. This is generally a computationally expensive process using traditional techniques due to the large number of design variables associated with topology optimization problem. However, combining the e‐cient methods of the non-gradient HCA algorithm with the decoupled approach for RBDO aims to reduce this burden. The topology optimization was without constraint in previous applications of the HCA method. To accommodate RTBO, a mechanism for a global constraint for maximum allowable displacement is developed. This paper details the methodology for the six-sigma design of structures using topology optimization.

A globally convergent method to accelerate topology optimization using on-the-fly model reduction

The design of multi-materials requires the determination of several variables: morphology, components and geometric parameters. Previously developed methods help designer making these choices separately because they always lie on a reduction in the number of design variables, as some of them are fixed at the beginning of the study. This paper proposes a method to carry out the simultaneous definition of all these variables in a preliminary design step. The beginning of this work consists in the formalisation of the multi-scale decomposition of the architecture materials. This outline is a guide for the use of databases of materials and architectures, whose combination enables the generation of multi-materials. Within the huge solution space resulting from this approach, a genetic algorithm allows to find optimised materials to fulfil a set of requirements with a limited number of calculations. An analytic example illustrates the efficiency of this method and shows it can provide designers with several propositions of materials.

In this study, the binary bat algorithm (BBA) for structural topology optimization is implemented. The problem is to find the stiffest structure using a certain amount of material and some constraints using the bit-array representation method. A new filtering algorithm is proposed to make BBA find designs with no separated objects, no checkerboard patterns, less unusable material, and higher structural performance. A volition penalty function for topology optimization is also proposed to accelerate the convergence toward the optimal design. The main effect of using the BBA lies in the fact that the BBA is able to handle a large number of design variables in comparison with other well-known metaheuristic algorithms. Based on the numerical results of four benchmark problems in structural topology optimization for minimum compliance, the following conclusions are made: (1) The BBA with the proposed filtering algorithm and penalty function are effective in solving large-scale numerical topology optimization problems (fine finite elements mesh). (2) The proposed algorithm produces solid-void designs without gray areas, which makes them practical solutions that are applicable in manufacturing.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

This paper presents an efficient and comprehensive MATLAB code to solve two-dimensional structural topology optimization problems, including minimum mean compliance, compliant mechanism synthesis and multi-load compliance problems. The Unsmooth Variational Topology Optimization (UNVARTOP) method, developed by the authors in a previous work, is used in the topology optimization code, based on the finite element method (FEM), to compute the sensitivity and update the topology. The paper also includes instructions to improve the bisection algorithm, modify the computation of the Lagrangian multiplier by using an Augmented Lagrangian to impose the constraint, implement heat conduction problems and extend the code to three-dimensional topology optimization problems. The code, intended for students and newcomers in topology optimization, is included as an appendix (Appendix A) and it can be downloaded from https://github.com/DanielYago together with supplementary material.

Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework